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A339207
Triangle read by rows T(n, k) = Sum_{h>=0} Bernoulli(h)*binomial(k+h-1, h)*abs(Stirling1(n, h+k))*n^h for n>=0 and 0<=k<=n.
3
1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -5, 0, 1, 0, 24, 0, -15, 0, 1, 0, 0, 238, 0, -35, 0, 1, 0, -3396, 0, 1281, 0, -70, 0, 1, 0, 0, -51508, 0, 4977, 0, -126, 0, 1, 0, 1706112, 0, -408700, 0, 15645, 0, -210, 0, 1, 0, 0, 35028576, 0, -2267320, 0, 42273, 0, -330, 0, 1
OFFSET
0,13
LINKS
René Gy, Bernoulli-Stirling Numbers, INTEGERS 20 (2020), #A67. See Table 1 p. 9.
FORMULA
T(n, k) = Sum_{h>=0} Bernoulli(h)*binomial(k+h-1, h)*abs(Stirling1(n, h+k))*n^h.
EXAMPLE
Triangle begins
1;
0 1;
0 0 1;
0 -1 0 1;
0 0 -5 0 1;
0 24 0 -15 0 1;
0 0 238 0 -35 0 1;
...
MATHEMATICA
T[0, 0] = 1; T[n_, k_] := Sum[BernoulliB[j] * Binomial[k + j - 1, j] * Abs[StirlingS1[n, k + j]] * n^j, {j, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 28 2020 *)
PROG
(PARI) T(n, k) = sum(h=0, n-k, bernfrac(h)*binomial(k+h-1, h)*abs(stirling(n, h+k, 1))*n^h);
(Magma)
function T(n, k)
if k eq n then return 1;
else return (&+[Binomial(k+j-1, j)*Bernoulli(j)*(-1)^j*StirlingFirst(n, k+j)*n^j: j in [0..n-k]]);
end if; return T; end function;
[T(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 21 2022
(SageMath)
def A339207(n, k):
if (k==n): return 1
else: return sum( binomial(k+j-1, j)*bernoulli(j)*stirling_number1(n, k+j)*n^j for j in (0..n-k) )
flatten([[A339207(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 21 2022
CROSSREFS
Cf. A027641/A027642 (Bernoulli), A007318 (binomial), A000254 (unsigned Stirling1).
Sequence in context: A132795 A277031 A085198 * A372959 A199916 A363041
KEYWORD
sign,tabl
AUTHOR
Michel Marcus, Nov 27 2020
STATUS
approved